Thermodynamics Hw#3

1. Chemical and Mechanical Engineering 2300 / Thermodynamics I
Solution to Homework Assignment 3 (Lectures 1 - 7)
Prof. Geoff Silcox
Chemical Engineering
University of Utah
Due Monday, 2014 September 22, by 17:00
To ensure that you receive full credit for your solutions, write out all equations in
symbolic form, give numerical values for all variables and constants in the equations, and
write answers to definitions or conceptual problems in complete sentences. Approximate
answers are rounded to one significant figure. Your answers need to be reported with
three.
Problem 1
During some expansion and compression processes in piston-cylinder devices, gases
satisfy the relationship PVn = C, where n and C are constants. Calculate the work done by
a gas when it expands from a state of 150 kPa and 0.03 m3 to a final volume of 0.2 m3 for
the case of n = 1.3. If the gas in question is helium, with initial temperature 300 K, how
much heat (kJ) is transferred to the gas during this process? Approximate answer: Qin = 4
kJ.
Solution

3. Problem 2
A piston-cylinder device initially contains 0.07 m3 of nitrogen gas at 130 kPa and 120C.
The nitrogen is now expanded to a pressure of 100 kPa polytropically with a polytropic
exponent whose value is equal to the ratio of heat capacities: k = cp/cv = 1.400. This is
called an isentropic expansion and it occurs when the process is reversible and adiabatic.
Determine the final temperature and the boundary work done during this process.
Approximate answer: 2 kJ.
Solution

4. Problem 3
Determine the boundary work done by a gas during an expansion process if the measured
pressure and volume values at various states are measured as 300 kPa, 1 L; 290 kPa, 1.1
L; 270 kPa, 1.2 L; 250 kPa, 1.4 L; 220 kPa, 1.7 L; and 200 kPa, 2 L. Approximate
answer: 0.2 kJ.
Solution
The work done by the system on the surroundings is determined numerically using the
approximation
2 5   
1 1 1
    
0 5 out i i i i
W PdV . P P V V  
1
i

and

5. 10 3m3 295 0 1 280 0 1 260 0 2 235 0 3 210 0 3 kPa•L 0 243 kJ
         

       
out 1L W . . . . . .
Problem 4
Is there a table in the text that gives the specific volume or density of dry air? (a) If you
can find such a table, state the values of the specific volume (m3/kg) and density (kg/m3)
at 25ºC and 1 atm pressure. (b) Use the ideal gas law to calculate v and  and compare
these values to those found (or not) in the table. Approximate answer: (b) v = 0.8 m3/kg.
Solution
a) There are no tables in the text that give the specific volume of air at these conditions.
It is more convenient to use the ideal gas law.
b) The density and specific volume of air, assuming it behaves as an ideal gas, are
P  M 
101.325 kPa 28.97 kg kmol kg R T 8.315 kJ 1.184 25 273.15 K m
  3
u
3
kmol K
v 1 0.8445 m
kg


 
 
    

 
The assumption of ideality at these conditions is an excellent approximation.
Problem 5
Why are the symbols U, KE, and PE used to denote the energy change during a
process while the work and heat transfer are represented as W and Q?
Solution
Internal energy U, potential energy PE, and kinetic energy KE are properties of the
system and changes in their values depend only on their end points: U = U2 - U1. W
and Q represent the amounts of energy transferred to the system by work and heat. They
are not properties. Their values depend on the path the process takes.
Problem 6
Please define the following terms, using mathematical equations if necessary, and answer
any associated questions. Be sure that all definitions are complete sentences.
1. adiabatic
2. isothermal

6. 3. reversible
4. isobaric
5. polytropic
6. efficiency of a compression process
7. efficiency of an expansion process
8. pure substance
9. compressed liquid
10. saturated mixture
11. superheated vapor
12. state postulate
13. Given a saturated mixture,
a. Are T and p independent, intensive properties? Why or why not?
b. Are T and v independent, intensive properties? Why or why not?
c. Are T and u independent, intensive properties? Why or why not?
14. The average atmospheric pressure at the Salt Lake City Airport is about 650 mm Hg.
At what temperature will water boil in SLC?
Solution
1. adiabatic. No heat crosses the boundary of a system during an adiabatic process.
2. isothermal. A system remains at a constant, uniform temperature during an
isothermal process.
3. reversible. A reversible process is one in which the system remains infinitesimally
close to equilibrium at all times. This is an idealization that is approximated by many
real processes.
4. isobaric. A system remains at a constant, uniform pressure during an isobaric
process.
5. polytropic. During a polytropic process, the pressure and volume of the system are
related by the equation pvn = constant where n is a constant. The value of n usually
lies between 1 and k.
6. efficiency of a compression process. Actual compression processes require more
work than reversible compression processes. We define an efficiency, , to account
for this:
W W
reversible , 0 < η < 1, compression
actual


7. efficiency of an expansion process. Actual expansion processes deliver less work
than reversible expansion processes. We define an efficiency, , to account for this:
, 0 < η < 1, expansion actual reversible W W
8. pure substance. A system consisting of a pure substance has a uniform chemical
composition.
9. compressed liquid. A pure liquid substance in a single-phase system at equilibrium is
called a compressed liquid if its temperature is below the critical temperature.
10. saturated mixture. A pure substance in a two-phase system at equilibrium is called a
saturated mixture.

7. 11. superheated vapor. A pure substance in a single-phase, gaseous system at equilibrium
is called a superheated vapor if its temperature is below the critical temperature.
12. state postulate. Two, independent, intensive properties are required to specify the
intensive state of a pure substance. This postulate is an experimental observation.
13. Given a saturated mixture,
a. Are T and p independent, intensive properties? Why or why not?
No, because in a saturated mixture, the pressure (temperature) cannot be
varied without also changing the temperature (pressure).
b. Are T and v independent, intensive properties? Why or why not?
Yes, because in a saturated mixture, the temperature and specific volume can
be varied independently by changing the quality.
c. Are T and u independent, intensive properties? Why or why not?
Yes, because in a saturated mixture, the temperature and specific internal
energy can be varied independently by changing the quality.
14. The average atmospheric pressure at the Salt Lake City Airport is about 650 mm Hg.
At what temperature will water boil in SLC?
We start by converting the atmospheric pressure from mm Hg to kPa.
650 mm Hg101.325 kPa 86.66 kPa
atm 760 mm Hg P  
Interpolate in Table A-5 to obtain the corresponding saturation temperature. This is the
boiling point of water when the atmospheric pressure is 650 mm Hg.

91.76 99.61 91.76 (86.66 75) 95.4 C
   
T
sat 100 
75 
Problem 7
What form does the energy balance take for an isolated system? Give the integrated and
rate forms of the balance.
Solution
For a closed system,
dE  Q   W and  E  Q 
W
dt
in out in out

8. For an isolated system, there is no energy transfer across the system boundary by work or
heat:
dE  0 and  E 
0
dt
These equations state that the energy of the system is constant because the closed system
is not interacting in any way with the surroundings.